MA/CSSE 474 Theory of Computation Bottom-up parsing Pumping - PDF document

MA/CSSE 474 Theory of Computation Bottom-up parsing Pumping Theorem for CFLs Recap: Going One Way Lemma : Each context-free language is accepted by some PDA. Proof (by construction): The idea: Let the stack do the work. Two approaches: • Top down • Bottom up

Top-down VS Bottom-up Approach Top-down Bottom-up left-to-right left-to-right Read the input string leftmost rightmost Derivation forward backward Order of derivation discovery Bottom-Up PDA The outline of M is: M = ({ p , q },  , V ,  , p , { q }), where  contains: ● The shift transitions: (( p , c ,  ), ( p , c )), for each c   . ● The reduce transitions: (( p ,  , ( s 1 s 2 … s n .) R ), ( p , X )), for each rule X  s 1 s 2 … s n . in G . Undoes an application of this rule. ● The finish-up transition: (( p ,  , S ), ( q ,  )). Top-down parser discovers a leftmost derivation of the input string (If any). Bottom-up parser discovers a rightmost derivation (in reverse order)

Bottom-Up PDA The idea: Let the stack keep track of what has been found. Discover a rightmost derivation in reverse order. (1) E  E + T Start with the string of terminals and attempt to (2) E  T "pull it back" (reduce) to S. (3) T  T  F (4) T  F (5) F  ( E ) (6) F  id Shift Transitions: (7) ( p , id,  ), ( p , id) Reduce Transitions: (8) ( p , (,  ), ( p , () (1) ( p ,  , T + E ), ( p , E ) (9) ( p , ),  ), ( p , )) (2) ( p ,  , T ), ( p , E ) Example: (10) ( p , +,  ), ( p , +) (3) ( p ,  , F  T ), ( p , T ) id + id * id (11) ( p ,  ,  ), ( p ,  ) (4) ( p ,  , F ), ( p , T ) (5) ( p ,  , ) E ( ), ( p , F ) When the right side of a production is (6) ( p ,  , id), ( p , F ) on the top of the stack, we can replace it by the left side of that production… …or not! That's where the nondeterminism comes in: choice between shift and reduce; choice between two reductions. Hidden during class, revealed later: Solution to bottom-up example A bottom-up parser is sometimes called a shift-reduce parser. Show how it works on id + id * id State stack remaining input transition to use p  id + id * id 7 p id + id * id 6 p F + id * id 4 p T + id * id 2 p E + id * id 10 p +E id * id 7 p id+E * id 6 p F+E * id 4 p T+E * id 11 p *T+E id 7 p id*T+E  6 p F*T+E  3 p T+E  1 p E  0 q   Note that the top of the stack is on the left. This is what I should have done in the class for sections 1 and 2 (and I did do it for section 3).

Acceptance by PDA  derived from CFG • Much more complex than the other direction. • Nonterminals in the grammar that we build from the PDA M are based on a combination of M's states and stack symbols. • It gets very messy. • Takes 9½ dense pages in the textbook (265-274). • I think we can use our limited course time better. How Many Context-Free Languages Are There? (we had a slide just like this for regular languages) Theorem: For any finite input alphabet Σ , there is a countably infinite number of CFLs over Σ . Proof: ● Upper bound: we can lexicographically enumerate all the CFGs. ● Lower bound: Each of {a}, {aa}, {aaa}, … is a CFL. The number of languages over Σ is uncountable. Thus there are more languages than there are context- free languages. So there must be some languages that are not context- free .

Languages That Are and Are Not Context-Free a * b * is regular. A n B n = { a n b n : n  0} is context-free but not regular. A n B n C n = { a n b n c n : n  0} is not context-free. We will show this soon. Is every regular language also context-free? Showing that L is Context-Free Techniques for showing that a language L is context-free: 1. Exhibit a CFG for L . 2. Exhibit a PDA for L . 3. Use the closure properties of context-free languages. Unfortunately, these are weaker than they are for regular languages. union, reverse, concatenation, Kleene star intersection of a CFL with a regular language NOT intersection, complement, set difference

CFL Pumping Theorem Show that L is Not Context-Free Recall the basis for the pumping theorem for regular languages: A DFSM M. If a string is longer than the number of M's states… Why would it be hard to use a PDA to show that long strings from a CFL can be pumped?

Some Tree Geometry Basics The height h of a tree is the length of the longest path from the root to any leaf. The branching factor b of a tree is the largest number of children associated with any node in the tree . Theorem: The length of the yield (concatenation of leaf nodes) of any tree T with height h and branching factor b is  b h . Shown in CSSE 230. A Review of Parse Trees A parse tree , (a.k.a. derivation tree ) derived from a grammar G = ( V ,  , R , S ), is a rooted, ordered tree in which: ● Every leaf node is labeled with an element of   {  }, ● The root node is labeled S , ● Every interior node is labeled with an element of N (i.e., V -  ), ● If m is a non-leaf node labeled X and the children of m (left-to-right on the tree) are labeled x 1 , x 2 , …, x n , then the rule X  x 1 x 2 … x n is in R .

From Grammars to Trees Given a context-free grammar G : ● Let n be the number of nonterminal symbols in G . ● Let b be the branching factor of G Suppose that a tree T is generated by G and no nonterminal appears more than once on any path from the root: The maximum height of T is: The maximum length of T ’s yield is: The Context-Free Pumping Theorem We use parse trees, not machines, as the basis for our argument. Let L = L(G), and let w  L. Let T be a parse tree for w such that has the smallest possible number of nodes among all trees based on a derivation of w from G. Suppose L(G) contains a string w such that | w| is greater than b n . Then its parse tree must look like (for some nonterminal X): X[1] is the lowest place in the tree for which this happens. I.e., there is no other X in the derivation of x from X[2].

The Context-Free Pumping Theorem Derivation of w There is another derivation in G : S  * uXz  * uxz , in which, at X[1], the nonrecursive rule that leads to x is used instead of the recursive one that leads to vXy. So uxz is also in L ( G ). The Context-Free Pumping Theorem There are infinitely many derivations in G , such as: S  * uXz  * uvXyz  * uvvXyyz  * uvvxyyz Those derivations produce the strings: uv 2 xy 2 z , uv 3 xy 3 z , uv 4 xy 4 z , … So all of those strings are also in L ( G ).

The Context-Free Pumping Theorem If rule 1 is X  X a , we could have v =  . If rule 1 is X  a X , we could have y =  . But it is not possible that both v and y are  . If they were, then the derivation S  * uXz  * uxz would also yield w and it would create a parse tree with fewer nodes. But that contradicts the assumption that we started with a parse tree for w with the smallest possible number of nodes. The Context-Free Pumping Theorem The height of the subtree rooted at [1] is at most: So | vxy |  .

The Context-Free Pumping Theorem Write it in contrapositive If L is a context-free language, then form. Try to  k  1 (  strings w  L , where | w |  k do this before (  u , v , x , y , z ( w = uvxyz , going on. vy   , | vxy |  k, and  q  0 ( uv q xy q z is in L )))). Pumping Theorem contrapositive • We want to write it in contrapositive form, so we can use it to show a language is NOT context-free. Original: If L is a context-free language, then  k  1 (  strings w  L , where | w |  k (  u , v , x , y , z ( w = uvxyz , vy   , | vxy |  k, and  q  0 ( uv q xy q z is in L )))). Contrapositive: If  k  1 (  string w  L , where | w |  k (  u , v , x , y , z ( w = uvxyz , vy   , | vxy |  k, and  q  0 ( uv q xy q z is not in L )))), then L is not a CFL.